3.021J / 1.021J / 10.333J / 18.361J / 22.00J Introduction ... · II. Atomistic and molecular...

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3.021J / 1.021J / 10.333J / 18.361J / 22.00J Introduction to Modeling and Simulation Markus Buehler, Spring 2008

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1

1.021/3.021/10.333/18.361/22.00 Introduction to Modeling and Simulation

Part II - lecture 8

Atomistic and molecular methods

2

Content overviewLectures 2-10February/March

Lectures 11-19March/April

Lectures 20-27April/May

I. Continuum methods1. Discrete modeling of simple physical systems:

Equilibrium, Dynamic, Eigenvalue problems2. Continuum modeling approaches, Weighted residual (Galerkin) methods,

Variational formulations3. Linear elasticity: Review of basic equations,

Weak formulation: the principle of virtual work, Numerical discretization: the finite element method

II. Atomistic and molecular methods1. Introduction to molecular dynamics2. Basic statistical mechanics, molecular dynamics, Monte Carlo3. Interatomic potentials4. Visualization, examples5. Thermodynamics as bridge between the scales6. Mechanical properties – how things fail7. Multi-scale modeling8. Biological systems (simulation in biophysics) – how proteins work and

how to model them

III. Quantum mechanical methods1. It’s A Quantum World: The Theory of Quantum Mechanics2. Quantum Mechanics: Practice Makes Perfect3. The Many-Body Problem: From Many-Body to Single-Particle4. Quantum modeling of materials5. From Atoms to Solids6. Basic properties of materials7. Advanced properties of materials8. What else can we do?

3

Overview: Material covered Lecture 1: Introduction to atomistic modeling (multi-scale modeling paradigm, difference between continuum and atomistic approach, case study: diffusion)

Lecture 2: Basic statistical mechanics (property calculation: microscopic states vs. macroscopic properties, ensembles, probability density and partition function, solution techniques: Monte Carlo and molecular dynamics)

Lecture 3: Basic molecular dynamics (advanced property calculation, chemical interactions)

Lecture 4: Interatomic potential and force field (pair potentials, fitting procedure, force calculation, multi-body potentials-metals/EAM & applications, neighbor lists, periodic BCs, how to apply BCs)

Lecture 5: Interatomic potential and force field (cont’d) (organic force fields, bond order force fields-chemical reactions, additional algorithms (NVT, NPT), application: mechanical properties –basic introduction)

Lecture 6: Application to mechanics of materials-brittle materials (significance of fractures/flaws, brittle versus ductile behavior [motivating example], basic deformation mechanisms in brittle fracture, theory, case study: supersonic fracture (example for model building); case study: fracture of silicon (hybrid model), modeling approaches: brittle-pair potential/ReaxF, applied to silicon

Lecture 7: Application to mechanics of materials-ductile materials (dislocation mechanisms), case study: failure of copper nanocrystal, supercomputing/parallelization

Lecture 8: Review session

Lecture 9: QUIZ

4

II. Atomistic and molecular methods

Lecture 8: Review session

Outline:1. Review of material covered in part II

1.1 Atomistic and molecular simulation algorithms1.2 Property calculation1.3 Potential/force field models1.4 Applications

2. Concluding remarks

Goal of today’s lecture: Review main concepts of atomistic and molecular dynamicsPrepare you for the quiz on Thursday

5

1.1 Atomistic and molecular simulation algorithms

1.2 Property calculation1.3 Potential/force field models

1.4 Applications

Goals: Basic MD algorithm (integration scheme), initial/boundary conditions, numerical issues (supercomputing)

6

Practice quiz II

7

Basic concept: atomistic methods

)(),(),( tatvtr iii

PDE solved in MD:i

ii rd

rdUdt

rdm )(2

2

−= }{ jrr =

Ni ..1=

Ni ..1=)( Fma =

Results of MD simulation:

vi(t), ai(t)ri(t)

xy

z

N particles

mass mi

8

Complete MD updating scheme

( ) ...)()(2)()( 20000 +Δ+Δ+Δ−−=Δ+ ttattrttrttr iiii

ijjij amf ,, =

Positions at t0

Accelerationsat t0

Positions at t0-Δt

jijij mfa /,, =

rx

Ff ijij

,, =

(4) Obtain force vectors from potential (sum over contributions from all neighbor of atom j)

Potential

(2) Updating method (integration scheme - Verlet)

(3) Obtain accelerations from forces

⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦⎤

⎢⎣⎡−⎥⎦

⎤⎢⎣⎡=

612

4)(rr

r σσεφ

(1) Initial conditions: Positions & velocities at t0 (random velocities so that initial temperature is represented)

rrF

d)(dφ

−=

Nj ..1=∀

9

Algorithm of force calculation

for i=1..N

for j=1..N (i ≠ j)

[add force contributions]

6 5

j=1

2 3

4

…

cutr

i

10

Summary: Atomistic simulation – numerical approach “molecular dynamics – MD”

Atomistic model; requires atomistic microstructure and atomic positions at beginning (initial conditions), initial velocities from random distribution according to specified temperatureStep through time by integration scheme (Verlet)Repeat force calculation of atomic forces based on their positionsExplicit notion of chemical bonds – captured in interatomic potential

Loop

Set particlepositions

Assign particlevelocities

Calculate forceon each particle

Move particles bytimestep Dt

Save current positions and

velocities

Reachedmax. number of

timesteps?

Stop simulation Analyze dataprint results

11

Practice quiz II

12

Force calculation for N particles: pseudocode

Requires two nested loops, first over all atoms, and then over all other atoms to determine the distance

for i=1..N:

for j=1..N (i ≠ j):

determine distance between i and j

calculate force and energy (if rij < rcut, cutoff radius)

add to total force vector / energy

time ~ N2: computational disaster

13

Strategies for more efficient computation

Two approaches

1. Neighbor lists: Store information about atoms in vicinity, calculated in an N2 effort, and keep information for 10..20 steps

2. Domain decomposition into bins: Decompose system into small bins; force calculation only between atoms in local neighboring bins

14

Periodic boundary conditionsPeriodic boundary conditions allows studying bulk properties (no free surfaces) with small number of particles (here: N=3), all particles are “connected”

Original cell surrounded by 26 image cells; image particles move in exactly the same way as original particles (8 in 2D)

Particle leaving box enters on other side with same velocity vector.

Figure by MIT OpenCourseWare. After Buehler.

15

Parallel computing – “supercomputers”

Supercomputers consist of a very large number of individual computing units (e.g. Central Processing Units, CPUs)

Images removed due to copyright restrictions

Photos of supercomputers.

16

Domain decomposition

Each piece worked on by one of the computers in the supercomputer

17

Modeling and simulation

18


Images removed due to copyright restrictions.

Actual map of Boston area subway lines compared to the simplified map found at mbta.com.

19


M

S

M

S

M

S

20

Atomistic versus continuum viewpoint

21

Diffusion: Phenomenological description

http://www.uic.edu/classes/phys/phys450/MARKO/N004.html

Ink dot

Tissue

Ink

conc

entra

tion

2

2

dxcdD

tc=

∂∂ 2nd Fick law (governing

equation)

BC: c (r = ∞) = 0IC: c (r=0, t = 0) = c0

Physical problem

Image removed due to copyright restrictions.

Please see http://www.uic.edu/classes/phys/phys450/MARKO/dif.gif.

http://www.uic.edu/classes/phys/phys450/MARKO/N004.html

22

Continuum model: Empirical parameters

Continuum model requires parameter that describes microscopic processes inside the material

Need experimental measurements to calibrate

Microscopicmechanisms

???

Continuummodel

Length scale

Time scale“Empirical”or experimentalparameterfeeding

23

Atomistic model of diffusion

Diffusion at macroscale (change of concentrations) is result of microscopic processes, random motion of particle

Atomistic model provides an alternative way to describe diffusion

Enables us to directly calculate the diffusion constant from thetrajectory of atoms

Follow trajectory of atoms and calculate how fast atoms leave their initial position

txpDΔΔ

−=2

Concept:follow this quantity overtime

24

Atomistic model of diffusion

2rΔ

Average squares of displacements of all particles

( )22 )0()(1)( ∑ =−=Δi

ii trtrN

tr

Particles Trajectories

Mean Squared Displacement function

i

ii rd

rdUdt

rdm )(2

2

−= }{ jrr = Ni ..1=

Images courtesy of the Center for Polymer Studies at Boston University. Used with permission.

25

2rΔ

Calculation of diffusion coefficient

1D=1, 2D=2, 3D=3

slope = D

( )22 )0()(1)( ∑ =−=Δi

ii trtrN

tr

Position of atom i at time t

Position of atom i at time t=0

( ))(lim21 2 tr

dtd

dD

tΔ=

∞→

Einstein equation

26

???

SummaryMolecular dynamics provides a powerful approach to relate the diffusion constant that appears in continuum models to atomistictrajectories

Outlines multi-scale approach: Feed parameters from atomistic simulations to continuum models

MD

Continuummodel

Length scale

Time scale“Empirical”or experimentalparameterfeeding

27



1.4 Applications

Goals: How to calculate “useful” properties from MD runs (temperature, pressure, RDF, VAF,..); significance of averaging; Monte Carlo schemes

28

Property calculation: IntroductionHave:

“microscopic information”

Want: Thermodynamical properties (temperature, pressure, ..)Transport properties (diffusivities, shear viscosity, ..)Material’s state (gas, liquid, solid)

)(),(),( tatvtr iii Ni ..1=

29

Micro-macro relation

)(131)(

1

2 tvmNk

tTN

iii

B∑=

=

)(tT

t

Which to pick?

Specific (individual) microscopic states are insufficient to relate to macroscopic properties

Courtesy of the Center for Polymer Studies at Boston University. Used with permission.

30

Ergodic hypothesis: significance of averaging

Ergodic hypothesis:

Ensemble (statistical) average = time average

All microstates are sampled with appropriate probability density over long time scales

∑∑==

=>=<>=<tA Nit

TimeEnsNiA

iAN

AAiAN ..1..1

)(1)(1

Monte CarloAverage over Monte Carlo steps

NO DYNAMICAL INFORMATION

MDAverage over time steps

DYNAMICAL INFORMATION

drdprprpAAp r∫ ∫>=< ),(),( ρ property must

be properlyaveraged

31

Monte Carlo schemeConcept: Find convenient way to solve the integral

Use idea of “random walk” to step through relevant microscopic states and thereby create proper weighting (visit states with higher probability density more often)

Monte Carlo schemes: Many applications (beyond what is discussed here; general method to solve complex integrations)

drdprprpAAp r∫ ∫>=< ),(),( ρ

32

Monte Carlo scheme: area calculation

Ω

∫Ω

Ω= dxfA )(

Method to carry out integration (illustrate general concept)

Want:

E.g.: Area of circle (value of π)

4

2dACπ

=4π

=CA 1=d

33

Monte Carlo schemeStep 1: Pick random point in Step 2: Accept/reject point based on criterion (e.g. if inside or outside of circle and if in area not yet counted)Step 3: If accepted, add to the total sum otherwise

Ω

1)( =ixf

ix

∫Ω

Ω= dxfAC )(

∑=i

iA

C xfN

A )(1

http://math.fullerton.edu/mathews/n2003/MonteCarloPiMod.html

AN : Attemptsmade

0)( =ixf

2/1=d

Courtesy of John H. Mathews. Used with permission.

http://math.fullerton.edu/mathews/n2003/MonteCarloPiMod.html

34

Area of Middlesex County (MSC)

∑=i

iA

SQMSC xfN

AA )(1

100 km

100

km 2km000,10=SQA

Fraction of points that lie within MS County

Expression provides area of MS County

x

y

Image courtesy of Wikimedia Commons.

35

Detailed view - schematic

NA=55 points (attempts)12 points within

22 km8.2181km22.010000)(1=⋅== ∑

ii

ASQMSC xf

NAA

12

36

Results U.S. Census Bureau

Taken from: http://quickfacts.census.gov/qfd/states/25/25017.html

2km8.2181=MSCAMonte Carlo result:

823.46 square miles = 2137 km2 (1 square mile=2.58998811 km2 )

http://quickfacts.census.gov/qfd/states/25/25017.html

37

Analysis of satellite images

200 m

Spy Pond (Cambridge/Arlington)

Image removed due to copyright restrictions.

Google Satellite image of Spy Pond, in Cambridge, MA.

38

Metropolis Hastings algorithm

drdprprpAAp r∫ ∫>=< ),(),( ρ

Images removed due to copyright restrictions. Images removed due to copyright restrictions.Please see: Fig. 2.7 in Buehler, Markus J. Atomistic Please see: Fig. 2.8 in Buehler, Markus J. AtomisticModeling of Materials Failure. New York, NY: Springer, 2008. Modeling of Materials Failure. New York, NY: Springer, 2008.

39

Molecular Dynamics vs. Monte Carlo

MD provides actual dynamical data for nonequilibrium processes (fracture, deformation, instabilities)

Can study onset of failure, instabilities

MC provides information about equilibrium properties(diffusivities, temperature, pressure)

Not suitable for processes like fracture

∑∑==

=>=<>=<tA Nit

TimeEnsNiA

iAN

AAiAN ..1..1

)(1)(1

40

Property calculation: Temperature and pressure

Temperature

Pressure( )>⋅−<

Ω= ∑

=

N

iiiii rfvmP

1

2

31

Distance vector multipliedby force vector (scalar product)

Volume

Kinetic contribution

><= ∑=

N

iii

B

vmNk

T1

2131

iii vvv ⋅=2

>⎟⎟⎠

⎞⎜⎜⎝

⎛⋅

∂∂

−Ω

=< ∑ ∑ ∑= = ≠=

=3..1 ..1 ,..1,

,, |)(21

31

i N aNrri

iii r

rr

rrvvmP

α ββαααα αβ

φ

41

Practice exam II

42

Formal approach: Radial distribution function (RDF)

ρρ /)()( rrg =

The radial distribution function is defined as

Provides information about the density of atoms at a given radius r; ρ(r) is the local density of atoms

ρ1

)()()(

2

2r

r

rrNrg

Δ

Δ

±Ω>±<

=

=drrrg 22)( π Number of particles that lie in a spherical shellof radius r and thickness dr

Local density

Density of atoms (volume)

Volume of this shell (dr)

Number of atoms in the interval 2rr Δ±

43

Radial distribution function: Solid versus liquid versus gas

Note: The first peak corresponds to the nearest neighbor shell, the second peak to the second nearest neighbor shell, etc.

In FCC: 12, 6, 24, and 12 in first four shells

5

4

3

2

1

00 0.2 0.4 0.6 0.8

g(r)

distance/nm

5

4

3

2

1

00 0.2 0.4 0.6 0.8

g(r)

distance/nm

Solid Argon

Liquid Argon

Liquid Ar(90 K)

Gaseous Ar(90 K)

Gaseous Ar(300 K)

Figure by MIT OpenCourseWare.

44

RDF and crystal structure

Peaks in RDF characterize NN distance, can infer from RDF about crystal structure

http://physchem.ox.ac.uk/~rkt/lectures/liqsolns/liquids.html

1st nearest neighbor (NN)

2nd NN3rd NN

…

5

4

3

2

1

00 0.2 0.4 0.6 0.8

g(r)

distance/nm

Solid Argon

Liquid Argon


http://physchem.ox.ac.uk/~rkt/lectures/liqsolns/liquids.html

45

Practice exam II

X

1st nearest neighbor (NN)

2nd NN3rd NN

copper: NN distance > 2 Å

not a liquid (sharp peaks)

46

Graphene/carbon nanotubes

Graphene/carbon nanotubes (rolled up graphene)NN: 1.42 Å, second NN 2.46 Å …

RDF

Images removed due to copyright restrictions. Please see:

http://weblogs3.nrc.nl/techno/wp-content/uploads/080424_Grafeen/Graphene_xyz.jpg

http://depts.washington.edu/polylab/images/cn1.jpg

http://depts.washington.edu/polylab/images/cn1.jpg



47

Practice quiz II

No

No

No

48

Velocity autocorrelation function

http://www.eng.buffalo.edu/~kofke/ce530/Lectures/Lecture12/sld010.htm

∑=

Δ+>=Δ+=<N

iii tntvtv

Ntnttvv

1000 )()(1)()0(

VAF

t

Damping due to dissipation - solid

Courtesy of the Department of Chemical and Biological Engineering of the University at Buffalo. Used with permission.

http://www.eng.buffalo.edu/~kofke/ce530/Lectures/Lecture12/sld010.htm

49

Velocity autocorrelation function: Solid, liquid, ideal gas

T

VAF

1

Liquid

Dense gas

Ideal gas

Solid


50

VAF used to calculate diffusion constant

')()0(31 '

0'

dttvvDt

t∫∞=

=

><=

Diffusion coefficient (see additional notes available on MIT Server“Additional notes re. velocity autocorrelation function”):

Diffusion coefficient can be obtained from both VAF and the MSD

51

Practice quiz II

52

Mean Squared Displacement function:transport of particles = diffusivities

( )2..1

2 )0()(1)( ∑=

=−>=Δ<Ni

ii trtrN

tr

Position of atom i at time t

Position of atom i at time t=0

Courtesy of Sid Yip. Used with permission.

53

Practice quiz II

No

No

No

X

X

X *

* Based on autocorrelationfunction (time dependent)

54

( )>⋅+<= ∑=

N

iiiii frvm

VP

1

2

31

><= ∑=

N

iii

B

vmNk

T1

2131

iii vvv ⋅=2

∑∑==

+>=<iN

kkiki

N

i i

ttvtvNN

tvv11

)()(11)()0(

>±Ω±

=<Δ

Δ

ρ)()()(

2

2r

r

rrNrg

( )22 )0()(1)( ∑ =−>=Δ<i

ii trtrN

tr

Temperature

Pressure

Stress

MSD

RDF

VAF

Direct

Direct

Direct

Diffusivity

Atomic structure (signature)

Diffusivity, phase state, transport properties

>⎟⎟⎠

⎞⎜⎜⎝

⎛⋅

∂∂

+−<Ω

= ∑ ∑≠

=α ββα

ααα αβ

φσa

rrji

jiij rrr

rruum

,,,, |)(

211

Property Definition Application

Summary – property calculation

55

Material properties: Classification

Structural – crystal structure, RDF, VAF

Thermodynamic -- equation of state, heat capacities, thermal expansion, free energies, use RDF, VAF, temperature, pressure

Mechanical -- elastic constants, cohesive and shear strength, elastic and plastic deformation, fracture toughness, use Virial Stress

Vibrational -- phonon dispersion curves, vibrational frequency spectrum, molecular spectroscopy, use VAF

Transport -- diffusion, viscous flow, thermal conduction, use MSD, temperature, VAF (viscosity, thermal conductivity)

56



1.4 Applications

Goals: How to model chemical interactions between particles (interatomic potential, force fields); applications to metals, proteins

57

Concept: Repulsion and attraction

“point particle” representation

Attraction: Formation of chemical bond by sharing of electronsRepulsion: Pauli exclusion (too many electrons in small volume)

r

Electrons

Core

e

Energy Ur

Repulsion

Attraction

1/r12 (or Exponential)

1/r6

Radius r (Distancebetween atoms)


58

Generic shape of interatomic potential

e

Energy Ur

Repulsion

Attraction

1/r12 (or Exponential)

1/r6

Radius r (Distancebetween atoms)

Attraction: Formation of chemical bond by sharing of electronsRepulsion: Pauli exclusion (too many electrons in small volume)


Harmonic oscillatorr0

φ

r~ k(r - r0)2

Effective interatomic potential


Many chemical bonds show this generic behavior

59

Atomic interactions – different types of chemical bonds

Primary bonds (“strong”)Ionic (ceramics, quartz, feldspar - rocks) Covalent (silicon) Metallic (copper, nickel, gold, silver)(high melting point, 1000-5,000K)

Secondary bonds (“weak”)Van der Waals (wax, low melting point) Hydrogen bonds (proteins, spider silk)(melting point 100-500K)

Ionic: Non-directional (point charges interacting)Covalent: Directional (bond angles, torsions matter)Metallic: Non-directional (electron gas concept)

Wea

ker b

ondi

ng

60

How to calculate forces from the “potential”or “force field”

Define interatomic potentials, that describe the energy of a set of atoms as a function of their coordinates

“Potential” = “force field”

)(rUU totaltotal =

NirUF totalri i..1)( =−∇=

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

∂∂

∂∂

=∇iii

r rrri,3,2,1

,,

Depends on position ofall other atoms

Change of potential energydue to change of position ofparticle i (“gradient”)

{ } Njrr j ..1==

Position vector of atom i

61

Practice quiz II

Pair potentialNote cutoff!

62

Pair potentials: energy calculation

Pair wise summation of bond energies

∑ ∑≠= =

=N

jii

N

jijrU

,1 121

tot )(φ

)( ijrφ

∑=

=N

jiji rU

1

)( φEnergy of atom i

Pair wiseinteraction potential

Simple approximation: Total energy is sum over the energy of all pairs of atoms in the system

=ijr distance betweenparticles i and j

avoid double counting

12 5

3 4

1 r12

2r25 5

3 4

63

Total energy of pair potential

Assumption: Total energy of system is expressed as sum of the energy due to pairs of atoms

( )32312321131221 φφφφφφ +++++= totalU

∑ ∑≠= =

=N

jii

N

jijtotal rU

,1 121 )(φ

)( ijij rφφ =with

beyond cutoff=0

=0

12r 23r

64

Force calculation – pair potential

ij

ij

rr

Fd

)(dφ−=

Forces can be calculated by taking derivatives from the potential function

Force magnitude: Negative derivative of potential energy with respect to atomic distance

To obtain force vector Fi, take projections into the three axial directions

x1

x2fij

ii r

xFf =

x1

x2

ijr

j

i

Component i of vector ijr

ijij rr =1f

2f

65

Force calculation – pair potential

Forces on atom 3: only interactionwith atom 2 (cutoff)

23

23

d)(d

rrF φ

−=23rxFf i

i = 2323 rr =ijr

1. Determine distance between atoms 3 and 2, r232. Obtain magnitude of force vector based on derivative of potential3. To obtain force vector Fi, take projections into the

three axial directionsComponent i of vector

66

Interatomic pair potentials: examples

Morse potential

Lennard-Jones 12:6 potential(excellent model for nobleGases, Ar, Ne, Xe..)

Buckingham potential

Harmonic approximation

( ) ( ))(exp2)(2exp)( 00 rrDrrDr ijijij −−−−−= ααφ

6

exp)( ⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛−=

ij

ijij r

Cr

Ar σσ

φ

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎟⎠

⎞⎜⎜⎝

⎛=

612

4)(ijij

ij rrr σσεφ

67

Lennard-Jones potential – example for copper

LJ potential – parameters for copper (Cleri et al., 1997)

ε

~σ

68

Derivative of LJ potential ~ force

Image removed due to copyright restrictions.Please see: Fig. 2.14 in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, 2008.

69

Determination of parameters for atomistic interactions

Often, parameters are determined so that the interatomic potential reproduces experimental results / quantum mechanical results (part III)

Example – elastic properties

'')(2

2

φφ=

∂∂

=r

rk

Calculate K as a function of ε and σ (for LJ potential):

Then find two (or more) properties (experimental, for example), that can be used to fit the LJ/BP parameters (e.g. Young’s modulus E = 120 GPa)

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛=

612

4)(rr

r σσεφ

Shear modulusYoung’s modulus

μ3/8=E))21(3/( ν−= EK4/1=ν

Vkr /2/20=μ

4/30aV =

3/64 σε=K

6

exp)( ⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛−=

rCrAr σ

σφ

70

Cutoff radius

∑=

=neighNj

iji rU..1

)(φ6 5

j=1

2 3

4

…

cutr

i φ

r

LJ 12:6potential

ε0

~σ

cutr

Cutoff radius = considering interactions only to a certain distanceBasis: Force contribution negligible (slope)

∑=

=N

jiji rU

1

)( φ

71

Derivative of LJ potential ~ force

Beyond cutoff: Changes in energy (and thus forces) small

rrF

d)(dφ

−=

)(rφ

cutr

potentialshift

force not zero

72

Crystal structure and potentialThe regular packing (ordering) of atoms into crystals is closely related to the potential detailsMany local minima for crystal structures exist, but materials tend to go to the structure that minimizes the energy; often this can be understood in terms of the energy per atomic bond and the equilibrium distance (at which a bond features the most potential energy)

φ

r

LJ 12:6potential

ε

~σ

2D example

Square lattice Hexagonal lattice

N=4 bonds N=6 bonds per atom

73

Example

time

http://polymer.bu.edu/java/java/LJ/index.html

Initial: cubic lattice Transforms into triangular lattice

Courtesy of the Center for Polymer Studies at Boston University. Used with permission.

http://polymer.bu.edu/java/java/LJ/index.html

74

Practice quiz II

75

All bonds are not the same: metals

© 2006 Markus J. Buehler, CEE/MIT

Pair potentials: All bonds Reality: Have environment are equal! effects; it matter that there is

a free surface!

76

Embedded-atom method (EAM)

)()(21

..1i

Njiji Fr

neigh

ρφφ += ∑=

Pair potential energy

Embedding energy

as a function of electron density

iρ Electron density at atom ibased on a pair potential:

∑=

=neighNj

iji r..1

)(πρ

new

Models other than EAM (alternatives):Glue model (Ercolessi, Tosatti, Parrinello)Finnis SinclairEquivalent crystal models (Smith and Banerjee)

77

Effective pair interactions: EAM potential

Can describe differences between bulk and surface

∑∑∑ +=≠ i

ijii j

ij FrU )()(21

,

ρφ

Pair potential energy Embedding energy

as a function of electron density

Embedding term: depends on environment, “multi-body”

EAM=Embedded atom method

Bulk

Surface

Effe

ctiv

e pa

ir po

tent

ial (

eV )

1

0.5

0

-0.53 42

r (A)o


78

Summary: EAM method

State of the art approach to model metalsVery good potentials available for Ni, Cu, Al since late 1990s, 2000sNumerically efficient, can treat billions of particlesNot much more expensive than pair potential, but describes physics much better

Recommended for use!

79

Model for chemical interactions

Similarly: Potentials for chemically complex materials – assume that total energy is the sum of the energy of different types of chemical bonds

bondHvdWMetallicCovalentElec −++++= UUUUUUtotal

80

Concept: energy landscape for chemically complex materials

Different energy contributions from different kinds of chemical bonds are summed up individually, independently

Implies that bond properties of covalent bonds are not affected by other bonds, e.g. vdW interactions, H-bonds

Force fields for organic substances are constructed based on this concept:

water, polymers, biopolymers, proteins …

bondHvdWMetallicCovalentElec −++++= UUUUUUtotal

Summary: CHARMM-type potential

Utotal = UElec +UCovalent +UMetallic +UvdW +UH−bond

81

UCovalent =Ustretch +Ubend +U rot

1φ ( )2stretch = k

2 stretch r − r0

1φ ( )2bend = k

2 bend θ −θ0

1φrot = krot (1− cos(ϑ))

UElec : Coulomb potentialqiqφ(r j

ij ) = ε1rij

UvdW : LJ potential⎡⎛ ⎞

12⎛ ⎞

6⎤σ σφ(r ) = 4ε ⎢⎜ ⎟ − ⎜ ⎟ ⎥ij ⎜⎢⎝ r ⎟ ⎜⎠ r ⎟

⎣ ij ⎝ ij ⎠ ⎥⎦

UH−bond :⎡ ⎛ ⎞

12⎛ ⎞

10⎤Rφ(r ) = ⎢5⎜ H−bond RD ⎟ − 6⎜ H−bond−

⎟ ⎥ 4ij H bond cos (θ⎜ ⎟ ⎜ ⎟⎢ r r ⎥

DHA )⎣ ⎝ ij ⎠ ⎝ ij ⎠ ⎦

1φrot = k (1 cos( ))2 rot − ϑ

2

82

Summary: CHARMM force field

Widely used and accepted model for protein structures

Programs such as NAMD have implemented the CHARMM force field

Problem set 2, GenePattern NAMD module, study of a protein domain part of human vimentin intermediate filaments

83

How can one accurately describe the transition energies during chemical reactions?

Use computationally more efficient descriptions than relying on purely quantum mechanical (QM) methods (see part III, methods limited to 100 atoms)

22

H

HCCHHCCH2

−=−→−≡−+

involves processeswith electrons

Reactive potentials ReaxFF

q q

q q

q

q

qq q

A

B

A--B

A

B

A--B

??

84

Bond order based energy landscape

Bond length

Bond order

Energy

Bond length

Energy

Bond order potentialAllows for a more general description of chemistryAll energy terms dependent on bond order

Conventional potential(e.g. LJ, Morse)

Pauling

85

ReaxFF: A reactive force field

underover

torsanglevalCoulombvdWaalsbondsystem

EE

EEEEEE

++

++++= ,

2-body

multi-body

3-body 4-body

Total energy is expressed as the sum of various terms describingindividual chemical bonds

All expressions in terms of bond order

All interactions calculated between ALL atoms in system…

No more atom typing: Atom type = chemical element

86

Mg – water interaction – ReaxFF MD simulation

87

Effect of Pt catalyst

MD simulation clearly proves the effect of the catalyst in greatly enhancing the reaction rate

It also leads to more controlled reaction conditions

5

4

3

2

1

0 0

( )

2

600 K with Pt

600 K without Pt

0.1 0.2 0.3 0.4 0.5 Time ns

Wat

er m

olec

ules

Number of H O molecules over time


88

•O2 close to Pt surface•Chemisorption of O2 (Pt-O-O)•Dissociation Pt-O and formation of Pt-O-H (stable)•Formation of Pt-O-H2 as another H2 approaches; thereby leads to water and H-O-O molecule

•Several water molecules interact via hydrogen bonds

H2O forms at the Pt (111) surface

1 2 3


Formation mechanism

89

Summary: ReaxFF force field

Widely used model to describe chemically complex materials and chemical reactions

Can describe covalent bonds, vdW bonds, H-bonds, electrostatic bonds, metallic bonds

Programs such as CMDF have implemented the ReaxFF force field

Problem set #2, GenePattern CMDF module, study of fracture of a silicon single crystal

90

Summary: force fields discussed in IM/S

Pair potentials (LJ, Morse, harmonic potentials, biharmonic potentials), used for single crystal elasticity (pset #1) and fracture model (pset #2)Pair potential

EAM (=Embedded Atom Method), used for nanowire (pset #1)Multi-body potential

ReaxFF (reactive force field), used for CMDF/fracture model of silicon (pset#2)Multi-body potential (bond order potential)

Tersoff (nonreactive force field), used for CMDF/fracture model of silicon (note: Tersoff only suitable for elasticity of silicon)Multi-body potential (bond order potential)

CHARMM force field (organic force field), used for protein simulations (pset#2)Multi-body potential (angles, for instance)

91

Practice quiz II

ReaxFF

EAM, LJ

CHARMM-type (organic)

ReaxFF

CHARMM

92

f

x

Steered Molecular Dynamics (SMD) mimics AFM single molecule experiments

xk

fAtomic force microscope

k xf

pset #2, problem 1

93

Comparison – CHARMM vs. ReaxFF

M. Buehler, JoMMS, 2007

Covalent bondsnever break in CHARMM

94



1.4 Applications

Goals: Select applications of MD to address questions in materials science (ductile versus brittle materials behavior); observe what can be done with MD

95

MD simulation: chemistry as bridge between disciplines

Images removed due to copyright restrictions.Please see: Fig. 2.1 in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, 2008.

vi(t), ai(t)

xy

z

N particlesmass mi

96

Application – protein folding

ACGT

Four letter code “DNA”

Combinationof 3 DNA letters equals a amino acid

E.g.: Proline –

CCT, CCC, CCA, CCG

Sequence of amino acids“polypeptide” (1D structure)

Transcription/translation

Folding (3D structure)

.. - Proline - Serine –Proline - Alanine - ..

Goal of protein folding simulations:Predict folded 3D structure based on polypeptide sequence

97

Movie: protein folding with CHARMM

de novo Folding of a Transmembrane fd Coat Proteinhttp://www.charmm-gui.org/?doc=gallery&id=23

Quality of predicted structures quite good

Confirmed by comparison of the MSD deviations of a room temperature ensemble of conformations from the replica-exchange simulations and experimental structures from both solid-state NMR in lipid bilayers and solution-phase NMR on the protein in micelles)

Polypeptide chain


Screenshots from protein folding video, which can be found here:http://www.charmm-gui.org/?doc=gallery&id=23.

http://www.charmm-gui.org/?doc=gallery&id=23

http://www.charmm-gui.org/?doc=gallery&id=23

98

Application – mechanics of materialstensile test of a wire

Brittle Ductile

Strain

Stre

ss

BrittleDuctile

Necking

Figures by MIT OpenCourseWare.

99

Practice quiz II

100

Ductile versus brittle materials

Difficultto deform,breaks easily

Easy to deformhard to break

BRITTLE DUCTILE

Glass Polymers Ice...

Shear load

Copper, Gold


101

“Macro, global”

“Micro (nano), local”

)(rσ r

Deformation of materials:Nothing is perfect, and flaws or cracks matter

Griffith, Irwine and others: Failure initiates at defects, such as cracks, or grain boundaries with reduced traction, nano-voids, other imperfections

Failure of materials initiates at cracks

102

Crack extension: brittle response


103

Basic fracture process: dissipation of elastic energy


104

Limiting speeds of cracks: linear elastic continuum theory

• Cracks can not exceed the limiting speed given by the corresponding wave speeds unless material behavior is nonlinear• Cracks that exceed limiting speed would produce energy (physically impossible - linear elastic continuum theory)

Images removed due to copyright restrictions.Please see: Fig. 6.56 and 6.90 in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, 2008.

105

Summary: mixed Hamiltonian approach

Tersoff

( )TersoffReaxFFReaxFFReaxFFTersoffReaxFF )1()( FwFxwF −+=−

ReaxFFw Tersoffw

xxwxw ∀=+ 1)()( TersoffReaxFF

Tersoff

xImage removed due to copyright restrictions.

See fig. 1 in Buehler, Markus J., Adri C. T. van Druin, and William Goddard III. "Multiparadigm Modeling of Dynamical Crack Propagation in Silicon Using a

Reactive Force Field." Phys Rev Lett 96 (2006): 095505.

106

Atomistic fracture mechanism

M.J. Buehler et al., Phys. Rev. Lett., 2007


Fig. 3 from Buehler, Markus, Harvey Tang, Adri C. T. van Duin, and William A. Goddard III. "Threshold Crack Speed Controls Dynamical Fracture of

Silicon Single Crystals." Phys Rev Letter 99 (2007): 165502.

50

Lattice shearing: ductile response

Instead of crack extension, induce shearing of atomic latticeDue to large shear stresses at crack tipLecture 7



τ

τ

τ

τ

Concept of dislocations

Localization of shear rather than homogeneous shear that requires cooperative motion

“size of single dislocation” = b, Burgers vector 54

additional half plane


τ

τ

τ

τ

109

A closer look: surface cracks

http://www2.ijs.si/~goran/sd96/e6sem1y.gif

Courtesy of Goran Drazic. Used with permission.

http://www2.ijs.si/~goran/sd96/e6sem1y.gif

110

Analysis of a one-billion atom simulation of work-hardening

Image removed due to copyright restrictions. Please see: Fig. 1c in Buehler, Markus J., et al. "The Dynamical Complexity of Work-hardening: A Large-scale Molecular Dynamics Simulation." Acta Mechanica Sinica 21 (2005): 103-111.

111

Final sessile structure – “brittle”

112

Practice quiz II

Copper, nickel (ductile) and glass, silicon (brittle)

For a material to be ductile, require dislocations, that is, shearing of lattices

Not possible here (disordered)

113

Practice quiz II


114

Suddenly stopping crack

Image removed due to copyright restrictions.Please see Fig. 6.57 and 6.74 in: Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, 2008.

115

2. Concluding remarks

116

Goals of part II - from lecture 1

You will be able to …

Carry out atomistic simulations of various processes (diffusion,deformation, fracture..)

Carbon nanotubes, nanowires, bulk metals, proteins, silicon crystals,…

Understand how to analyze atomistic simulations

Understand how to visualize (see snapshots on previous slide)

Understand how to link atomistic simulation results with continuum models

117

Important terminology and concepts

Force field Potential / interatomic potentialReactive force field / nonreactive force fieldChemical bonding (covalent, metallic, ionic, H-bonds, vdW..)Time stepContinuum vs. atomistic viewpointsMonte Carlo vs. Molecular DynamicsProperty calculation (temperature, pressure, diffusivity [transport properties]-VAF,MSD, structural analysis-RDF)Choice of potential for different materials (pair potentials, chemical complexity)MD algorithm: how to calculate forces, energies Applications: brittle vs. ductile, data analysis

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